Problem Set 4
Stochastic Foundations for Finance
Instructions: This problem set is due on 11/27 at 11:59 pm CST and is an individual assignment. All problems must be handwritten. Scan your work and submit a PDF file.
Problem 1 The dynamics of X are defined by \mathop{}\!\mathrm{d}X = \sigma \mathop{}\!\mathrm{d}z. Show that \operatorname{E}(X_{T}) = X_{0}, and hence that X is a martingale.
Problem 2 Consider two assets whose price processes are given by \frac{\mathop{}\!\mathrm{d}\mathbf{S}}{\mathbf{S}} = \boldsymbol{\mu} \mathop{}\!\mathrm{d}t + \boldsymbol{\sigma} \mathop{}\!\mathrm{d}\mathbf{z}, where \mathop{}\!\mathrm{d}\mathbf{z} is a vector of three independent Brownian motions z_{1}, z_{2}, and z_{3}. You know that \boldsymbol{\sigma} = \begin{pmatrix} 0.4 & 0.3 & -0.2 \\ 0.5 & -0.2 & -0.1 \end{pmatrix}.
- Compute the instantaneous correlation between the returns of each asset.
- Find a Brownian motion z_{4} = a_{1} z_{1} + a_{2} z_{2} + a_{3} z_{3} whose increments are independent from the instantaneous returns of the two assets.
Problem 3 Consider a stochastic discount factor in continuous time given by \frac{\mathop{}\!\mathrm{d}\Lambda}{\Lambda} = - r \mathop{}\!\mathrm{d}t - \lambda_{1} \mathop{}\!\mathrm{d}z_{1} - \lambda_{2} \mathop{}\!\mathrm{d}z_{2}, where z_{1} and z_{2} are independent Brownian motions. Suppose that you have two non-dividend paying assets with the following dynamics: \frac{\mathop{}\!\mathrm{d}S_{1}}{S_{1}} = \mu_{1} \mathop{}\!\mathrm{d}t + \sigma_{11} \mathop{}\!\mathrm{d}z_{1} + \sigma_{12} \mathop{}\!\mathrm{d}z_{2}, and \frac{\mathop{}\!\mathrm{d}S_{2}}{S_{2}} = \mu_{2} \mathop{}\!\mathrm{d}t + \sigma_{21} \mathop{}\!\mathrm{d}z_{1} + \sigma_{22} \mathop{}\!\mathrm{d}z_{2}. Suppose that r = 0.05, \sigma_{11} = 0.5, \sigma_{12} = 0.2, \sigma_{21} = -0.1, and \sigma_{22} = 0.4. Furthermore, you know that \lambda_{1} = 0.3 and \lambda_{2} = 0.5.
- Compute the instantaneous correlation between the returns of each asset.
- Compute \mu_{1} and \mu_{2}.
Problem 4 Suppose that the sales team of a trading desk just sold a European call option contract, i.e., over a 100 shares, to an important client. The contract is written on a non-dividend paying stock that trades for $210, expires in two years and has a strike price of $215. The risk-free rate is 6% per year with continuous compounding. A trader of the desk estimate that the volatility of the stock returns is 45% and expected to remain constant for the life of the contract.
- How many shares of the stock does the trader need to buy/sell initially in order to hedge the exposure created by the sale of the contract?
- How many risk-free bonds with face value $215 and expiring in two years does the trader need to buy/sell in order to make sure that the strategy is self-financing?
- Compute the implicit leverage of call defined as the amount invested in the stock (S C_{S}) over the price of the call (C).
Problem 5 Consider a European call option expiring in 6 months and with strike price equal to $42 on a non-dividend paying stock that currently trades for $40. Interestingly, the volatility of the stock is zero. If the risk-free rate is 6% per year with continuous compounding, what is the price of the option?